On the operad of associative algebras with...

Georgian Math. J. (2010), 26 pagesDOI 10.1515/GMJ.2010.010 © de Gruyter 2010

On the operad of associative algebraswith derivation

Jean-Louis Loday

Abstract. We study the operad of associative algebras equipped with a derivation. Weshow that it is determined by polynomials in several variables and substitution. Replac-ing polynomials by rational functions gives an operad which is isomorphic to the operadof “moulds”. It provides an efficient environment for doing integro-differential calculus.Interesting variations are obtained by using formal group laws. The preceding case corre-sponds to the additive formal group law. We unravel the notion of homotopy associativealgebra with derivation in the spirit of Kadeishvili’s work.

Keywords. Derivation, operad, dendriform, tridendriform, formal group law, Ito integral,mould.

2010 Mathematics Subject Classification. 17A30, 18D50, 18G55, 16S99.

Dedicated to Tornike Kadeishvili in honor of his sixtieth birthday

Introduction

To any one-dimensional formal group law we show how to associate a type ofalgebras, that is an operad, whose space of n-ary operations is the algebra of ra-tional functions in n variables. If the formal group law is polynomial, then thespaces of polynomials in n variables form a suboperad. We show that, for theadditive formal group law, the polynomial operad is simply the operad AsDer ofassociative algebras equipped with a derivation, and the rational functions operadRatFct is isomorphic to the operads of “moulds”. Since the operad RatFct containsboth the derivation operator and its inverse, that is the integral operator, the wholeintegro-differential calculus can be written in terms of operadic calculus. The op-erad RatFct is related to the dendriform and tridendriform operads, and severalothers.

In order to deal with formal group laws in noncommutative variables, one has tomodify the notion of operad and work with the “pre-shuffle algebras” introducedby M. Ronco in [17]. The idea is essentially to forget parallel composition in thedefinition of an operad by means of the partial compositions.

2 J.-L. Loday

In the appendix we show that the operad AsDer, which is quadratic, is a Koszuloperad. We describe explicitly the notion of “homotopy associative algebra withderivation”, that is AsDer1-algebra.Notation. In this paper K is a commutative unital algebra and all modules over Kare supposed to be free. Its unit is denoted by 1. Since most of the time we thinkof K as being a field we will say a vector space or a space for a free module over K.The tensor product of vector spaces over K is denoted by ˝. The tensor productof n copies of the space V is denoted by V ˝n. The symmetric group Sn acts onV ˝n by place-permutation.

The polynomial algebra, resp. the algebra of rational functions, on the set ofvariables ¹x1; : : : ; xnº is denoted by KŒx1; : : : ; xn�, resp. K.x1; : : : ; xn/. The unitis denoted by 1n or 1 if there is no confusion.

1 Prerequisites on operads

This is a very brief introduction on algebraic operads, whose purpose is essentiallyto set up notations. One can consult for instance [14], or [13], for more details.

1.1 Nonsymmetric operad

A nonsymmetric operad, or an ns operad for short, is a graded vector space¹Pnºn�0 equipped with a particular element of P1 denoted id and called the iden-tity operation, and a family of linear maps

�i1;:::;ik W Pk ˝Pi1 ˝ � � � ˝Pik ! Pn; n D i1 C � � � C ik;

which satisfy the following associative and unital properties. On the functor P WVect ! Vect defined as P .V / WD

LnPn ˝ V ˝n the operations �i1;:::;ik induce

a transformation of functors � W P ı P ! P . Then, � is supposed to be associa-tive. The element id can be interpreted as a morphism from the identity functor Ito P that we denote by � W I! P . Then, � is supposed to be a unit for � .

There is an alternative definition of a ns operad which uses the so-called partialcompositions. For any i � m the partial composition

ıi W Pm ˝Pn ! PmCn�1

evaluated on .�; �/ is the operation �1;:::;1;n;1;:::;1 evaluated on .�I id; : : : id; �;id; : : : ; id/. Pictorially it is represented by the following grafting of trees, where

On the operad of associative algebras with derivation 3

the root of � is grafted onto the i th leaf of �:

�

��

��

i

�

��

��

��

An ns operad can be defined as a graded vector space equipped with an identityoperation and partial compositions which satisfy the unital axioms and the fol-lowing two axioms (which replace associativity). For any � 2 Pl ; � 2 Pm and� 2 Pn:

Axiom I (sequential composition):

.� ıi �/ ıi�1Cj � D � ıi .� ıj �/; 1 � i � l; 1 � j � m:

It corresponds to the two possibilities of composing in the following diagram:

�

��

��

j

�

��

��

��

��

i

�

��

��

��

��

Axiom II (parallel composition):

.� ıi �/ ıkCm�1 � D .� ık �/ ıi �; 1 � i < k � l:

4 J.-L. Loday

It corresponds to the two possibilities of composing in the following diagram:

�

��

��

i k

�

��

��

��

��

1.2 Symmetric operads

A symmetric operad, or an operad for short, is a family of right Sn-modules¹P .n/ºn�0 equipped with transformation of functors � W P ı P ! P and� W I ! P which are associative and unital. Here P stands for the so-calledSchur functor defined as

P .V / WDM

n

P .n/˝Sn V ˝n:

Any operad and any ns operad gives rise to a notion of algebras over this operad.There is a definition of operad using the partial compositions. See loc. cit. fordetails.

2 Associative algebras with a derivation

2.1 Definition

Let A be a nonunital associative algebra over K. A derivation of A is a linear mapDA W A! A which satisfies the Leibniz relation

DA.ab/ D DA.a/b C aDA.b/

for any a; b 2 A. Let .A0;DA0/ be another associative algebra with a derivation.A morphism f W .A;DA/ ! .A0;DA0/ is a linear map f W A ! A0 which isa morphism of associative algebras and which commutes with the derivations:

f ıDA D DA0 ı f:


The operad governing the category of associative algebras with derivation ad-mits the following presentation. There are two generating operations, one of arity 1that we denote by D, and one of arity 2 that we denote by �. The relations are:

´� ı .�; id/ D � ı .id; �/;D ı � D � ı .D; id/C � ı .id;D/:

They account for the associativity of the product and for the Leibniz relation.Since, in the relations, the variables stay in the same order, the category of as-sociative algebras with derivation can be encoded by a nonsymmetric operad thatwe denote by AsDer. So it is determined by a certain family of vector spacesAsDern; n � 1, and composition maps

�i1;:::;ik W AsDerk˝AsDeri1 ˝ � � � ˝ AsDerik ! AsDern;

where n D i1 C � � � C ik .

2.2 Theorem. As a vector space AsDern is isomorphic to the space of polynomials Note 1Theorem2.2 and6.2 havethe samelabel.Pleasecheck.

in n variables:

AsDern D KŒx1; : : : ; xn�:

The composition map � D �i1;:::;ik is given by

�.P IQ1; : : : ;Qk/.x1; : : : ; xn/D P.x1 C � � � C xi1 ; xi1C1 C � � � C xi1Ci2 ; xi1Ci2C1 C � � � ; : : :/�Q1.x1; : : : ; xi1/Q2.xi1C1; : : :/ � � � :

Under this identification the operations id;D; � correspond to 11; x1 2 KŒx1� andto 12 2 KŒx1; x2� respectively. More generally the operation

.a1; : : : ; an/ 7! Dj1.a1/Dj2.a2/ � � �Djn.an/

corresponds to the monomial xj11 xj22 � � � x

jnn .

Graphically the operation xj11 xj22 � � � x

jnn is pictured as a planar decorated tree

as follows:

6 J.-L. Loday

� � �

Dj1 Dj2 � � � Djn

��

��

��

��

��

��

��

��

��

��

�n

Proof. Since � is an associative operation the space of n-ary operations generatedby � is one-dimensional. Let us denote by �n the composition of n � 1 copiesof �. Using the Leibniz relation we see that any composite of copies of � and Dcan be written uniquely as composites of copies of D first and then composites ofcopies of �. Hence any n-ary operation is a linear combination of operations ofthe form

�n ı .Dj1 ; : : : ;Djn/

for some sequence of nonnegative integers ¹j1; : : : ; jnº. We denote this operationby xj11 x

j22 � � � x

jnn . We obtain: AsDern D KŒx1; : : : ; xn�.

In order to prove the formula for the composition map � , it is sufficient to provethat

.P ıi Q/.x1; : : : ; xnCm�1/D P.x1; : : : ; xi�1; xi C � � � xiCm�1; xiCm; : : : ; xnCm�1/Q.xi ; : : : ; xiCm�1/;

where � ıi � is the i th partial composition, P 2 AsDern; Q 2 AsDerm. It issufficient to prove this formula when P and Q are monomials:


Dk1 � � � Dkm

�m

��

Di1 � � � Dji � � � Djn

�n

��

��

By direct inspection we see that it is sufficient to treat the case Q D �m,and in fact the case P D D`;Q D �m. In other words we need to computethe element D`.a1 � � � am/. By the Leibniz relation, for ` D 1, this is exactlythe action of the operation x1 C � � � C xm. Recursively we get that the operation.a1; : : : ; am/ 7! D`.a1 � � � am/ is .x1C� � �Cxm/` as expected. So we are done.

2.3 Lemma. The two binary operations a ` b and a a b of AsDer2 given by x1and x2 respectively satisfy the following relations

.a ` b/ a c D a ` .b a c/;.a a b/ ` c C .a a b/ a c D a a .b ` c/C a ` .b ` c/:

Proof. By a straightforward operadic calculus we get

x1 ı1 x1 D .x1 C x2/x1; x1 ı1 x2 D .x1 C x2/x2;x2 ı1 x1 D x3x1; x2 ı1 x2 D x3x2;x1 ı2 x1 D x1x2; x1 ı2 x2 D x1x3;x2 ı2 x1 D .x2 C x3/x2; x2 ı2 x2 D .x2 C x3/x3:

It follows immediately that

cx2 ı1 x1 D x1 ı2 x2;x1 ı1 x2 C x2 ı1 x2 D x1 ı2 x1 C x2 ı2 x1:

If we put a ` b WD x1.a; b/ and a a b WD x2.a; b/, then we get the expectedformulas.

8 J.-L. Loday

2.4 Remarks

The relations of Lemma 2.3 give rise to a new type of ns operad generated bytwo operations. Other examples include magmatic, dendriform [9], cubical [11],duplicial [10], compatible-two-associative [4].

Since AsDer is a ns operad, it is completely determined by its free algebraover one generator. This free algebra has also been computed in [7] by Guo andKeigher.

2.5 An elementary example

Let K D kŒy� be the polynomial algebra in one variable over the field k. Weconsider the associative algebra with derivation .A;DA/ D .kŒy�Œx�; @

@x/. In

AsDer.kŒy�/^ WDQn AsDern we consider the operation

exp.yD/ WDX

n�0

�yn

nŠDn

�

:

Then for any polynomial p.x/ 2 kŒx� the following formula holds in kŒy�Œx� DkŒx; y�:

exp

�

y@

@x

�

.p.x// D p.x C y/:

This is a key formula for studying vertex algebras, see for example [16].

3 First variation and dendriform algebras

Since the composition � in the operad AsDer needs only the sum, the product andthe substitution of variables to be defined, it can be extended to many generaliza-tions of the polynomial algebras: C1-functions, rational functions, etc., providedthat they are commutative. The rational function case is interesting since it permitsus to treat integration, represented by the rational function 11

x1, which is the inverse

of the operation x1 coding for derivation.

3.1 The rational functions operad

We define a new ns operad RatFct by

RatFctn WD K.x1; : : : ; xn/


(rational functions in the variables x1; : : : ; xn). The composition � is given by theformula of Theorem 6.2, or, equivalently by the partial composition formula:

.P ıiQ/.x1; : : : ; xnCm�1/WD P.x1; : : : ; xi�1; xi C � � � C xiCm�1; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

3.2 Proposition. The partial compositions � ıi � as defined above make RatFctinto a ns operad.

Proof. There are two axioms to check, cf. Section 1.1. The first one follows fromthe fact that addition of variables is a formal group law, that is F.x; y/ WD x C ysatisfies F.F.x; y/; z/ D F.x; F.y; z//. The second axiom follows from the factthat the algebra of rational functions is commutative.

We observe that the structure of associative algebra of RatFct1 is precisely thealgebra structure of the rational functions in one variable K.x1/.

3.3 Integro-differential calculus

In the operad RatFct the derivation operation D, represented by x1 2 K.x1/ DRatFct1, admits an inverse for composition, that is 1=x1, which is the integrationoperation

R. So we can write the integro-differential calculus within the operad

RatFct. Here is an example.For integrable functions f and g on R define

.f � g/.x/ WD f .x/Z x

0

g.t/dt and .f � g/.x/ WD�Z x

0

f .t/dt

�

g.x/:

Then, it is shown in [9] that these two operations satisfy the dendriform axioms(see below) as a consequence of integration by parts:

Z x

0

g.t/dt

Z x

0

h.t/dt DZ x

0

g.t/

�Z t

0

h.s/ds

�

dt CZ x

0

�Z t

0

g.s/ds

�

h.t/dt:

We should be able to recover this property by computing in the operad RatFct sincef � g D .1=x1/.f; g/ and f � g D .1=x2/.f; g/, for 1=x1; 1=x2 2 RatFct2.This is the object of Proposition 3.5.

More analogous formulas can be found in [3].

10 J.-L. Loday

3.4 Dendriform algebras

Let us recall from [8, 9] that a dendriform algebra is a vector space equipped withtwo binary operations a � b and a � b satisfying the relations

8


It follows immediately that

1

x1ı1

1

x1C 1x1ı1

1

x2D 1x1 C x2

1

x1C 1x1 C x2

1

x2D 1x1

1

x2D 1x1ı2

1

x1;

1

x2ı1

1

x1D 1x3

1

x1D 1x1ı2

1

x2;

1

x2ı1

1

x2D 1x3

1

x2D 1x2 C x3

1

x2C 1x2 C x3

1

x3

D 1x2ı2

1

x1C 1x2ı2

1

x2:

If we put a � b WD 1x1.a; b/ and a � b WD 1

x2.a; b/, then we get the expected

formulas.

As for the second assertion, we first prove the formula for the grafting of trees.The proof is a straightforward operadic calculus, whose steps are the following.In Dend.K/ we have s _ t D s � Y � t D .s � Y/ � t . We first computes � Y D �.�I s; Y/ D .� ı1s/ ıpC1 Y. Applying ' we get

'.s � Y/ D�12

x1ı1 s

�ıpC1 Y

D 1x1 C � � � C xp

'.s/.x1; : : : ; xp/ ın Y

D 1x1 C � � � C xp

'.s/.x1; : : : ; xp/:

A similar computation for the left product leads to the expected formula.

The rational functions '.t/ for t 2 PBTn are linearly independent (proof byinduction). Hence ' W Dendn D KŒPBTnC1�! RatFctn is injective for all n.

3.6 Examples

Here is the image of the pb trees under ' in low dimension:

1x1

1x2

��

12 J.-L. Loday

1x1

1x1Cx2

��

��

��

��

��

��

1x1Cx2

1x2

��

��

��

��

�1x1

1x3

1x2

1x2Cx3

��

��

1

x2Cx31x3

3.7 Associativity of �

Let us adopt the notation ˆ.u; v/ D 1uC 1

v. For any variables x1; x2; x3 the

formulaˆ.x1; x2/ˆ.x1 C x2; x3/ D ˆ.x2; x3/ˆ.x1; x2 C x3/

is immediate to satisfy (cf. the proof of Proposition 3.5):

�1

x1C 1x2

��1

x1 C x2C 1x3

�

D x1 C x2 C x3x1x2x3

D�1

x2C 1x3

��1

x1C 1x2 C x3

�

:

Viewed as an equality in the space RatFct3 of ternary operations, it simply saysthat the operation D� C � is associative. Indeed the left part of the equality isˆ ı1 ˆ and the right part is ˆ ı2 ˆ.

This formula is reminiscent of the cocycle condition of a 3-cochain � in groupcohomology:

�.x; y/C �.xy; z/ D x�.y; z/C �.x; yz/;

and to the construction of the McLane invariant of a crossed module.

3.8 Comparison with the operad of moulds

The work of J. Ecalle led F. Chapoton to introduce the operad of moulds in [2, 3].It is a nonsymmetric operad denoted by Mould, which is determined by

Mouldn WD K.x1; : : : ; xn/;


and by the partial compositions given by the formula

.P ıiQ/.x1; : : : ; xnCm�1/ WD .xi C � � � C xiCm�1/� P.x1; : : : ; xi�1; xi C � � � C xiCm�1; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

3.9 Proposition. There is an isomorphism of ns operads RatFract Š Mould.

Proof. By direct inspection we verify that the map RatFractn ! Mouldn given by

P.x1; : : : ; xn/ 7! .x1 C � � � C xn/P.x1; : : : ; xn/

is compatible with the operadic compositions. Since we are working with rationalfunctions, the element x1C� � �Cxn is invertible and so this map is an isomorphism.

3.10 Remarks

(a) Under the isomorphism of Proposition 3.9, Proposition 3.5 can be foundin [2].

(b) Observe that the polynomials do not form a suboperad of Mould.

(c) If we think of the operation 12 2 K.x1; x2/ D RatFct2 as a third binaryoperation, then it is easy to check that the three operations ¹�º D 1

x1;

¹�º D 1x2; ¹�º D 12 satisfy the 7 axioms of a graded tridendriform alge-

bra (cf. [1, 12]). We come back to this point in the next section.

4 Second variation of AsDer

In probability theory there is a variation of the integration by parts called the Ito in-tegral. Its counterpart in the derivation framework, that could be called Ito deriva-tion, is a linear map DA W A! A which satisfies the following relation:

DA.ab/ D DA.a/b C aDA.b/CDA.a/DA.b/:

One can treat both the derivation case, the Ito derivation case (and even more)by introducing a parameter � 2 K as follows (we could also work with a formalparameter q, that is take K D kŒq�).

14 J.-L. Loday

4.1 Parametrized derivation

By definition, a �-derivation is a linear map DA W A! A, where A is an associa-tive algebra, such that

DA.ab/ D DA.a/b C aDA.b/C �DA.a/DA.b/

for any a; b 2 A. For � D 0 we get the derivation, for � D 1 we get the Itoderivation. By convention � D 1 stands for the case where DA is an algebrahomomorphim: DA.ab/ D DA.a/DA.b/.

We denote by �- AsDer the operad of associative algebras equipped with a �-derivation.

We introduce the notation

�.x1; : : : ; xn/

WD .x1 C � � � C xn/C � � � C �k�1k.x1; : : : ; xn/C � � � C �n�1.x1 : : : xn/

where k.x1; : : : ; xn/ is the kth symmetric function of the variables x1; : : : ; xn.

4.2 Theorem. As a vector space �- AsDern is isomorphic to the space of polyno-mials in n variables:

�- AsDern D KŒx1; : : : ; xn�:

The composition map � is given by

�.P IQ1; : : : ;Qk/.x1; : : : ; xn/

D P.�.x1; : : : ; xi1/; �.xi1C1; : : : ; xi1Ci2/; � � � /�Q1.x1; : : : xi1/Q2.xi1Ci2 ; : : :/ � � � :

Under this identification the operations id;D; � correspond to 11; x1 2 KŒx1� andto 12 2 KŒx1; x2� respectively. More generally the operation

.a1; : : : ; an/ 7! Dj1.a1/Dj2.a2/ � � �Djn.an/

corresponds to the monomial xj11 xj22 � � � x

jnn .


Proof. The proof is the same as in the case � D 0 performed in the first section.See also the proof of Proposition 4.8.

4.3 Remark

The associativity property of the composition in the operad �- AsDer implies that

�.x1; : : : ; xi ; �.xiC1; : : : ; xiCj /; xiCjC1; : : : ; xn/ D �.x1; : : : ; xn/:

This formula can also be proved directly by observing that

1C ��.x1; : : : ; xm/ DmY

iD1.1C �xi /:

4.4 The parametrized operad �- RatFct

As in Section 2 we can put an operad structure on the rational functions by usingthe formulas of Theorem 4.2. It gives a new operad, denoted �- RatFct, for which

�- RatFctn D K.x1; : : : ; xn/;

and the partial composition is given by

.P ıiQ/.x1; : : : ; xnCm�1/

WD P.x1; : : : ; xi�1; �.xi ; � � � ; xiCm�1/; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

4.5 �-TriDendriform algebras

In [12] we introduced the notion of tridendriform algebra, which is an algebrawith 3 binary operations satisfying 7 relations (one for each of the cells of a tri-angle). The graded version was studied by Chapoton in [1]. There exists a para-metrized version which handles both versions (and more) as follows. By def-inition, a �-tridendriform algebra has 3 binary operations denoted by a � b;

16 J.-L. Loday

a � b; a � b and 7 relations (one for each cell of the triangle):8


Let F be any formal group law. By induction we define

F.x1; : : : ; xn/ WD F.F.x1; : : : ; xn�1/; xn/:

4.8 Proposition. For any formal group law F there is a well-defined nonsymmetricoperad RatFctF given by:

RatFctFn WD K.x1; : : : ; xn/;

.P ıiQ/.x1; : : : ; xnCm�1/WD P.x1; : : : ; xi�1; F .xi ; � � � ; xiCm�1/; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

Proof. Axiom I is an immediate consequence of the equality

F.x1; : : : ; xi ; F .xiC1; : : : ; xiCj /; xiCjC1; : : : ; xn/ D F.x1; : : : ; xn/;

which follows from the associativity of F . Axiom II is a consequence of thecommutativity of the algebra of rational functions.

5 P -algebras with derivation

Let P be an algebraic operad, whose space of n-ary operations is denoted byP .n/. Recall that P .n/ is a right Sn-module. A derivation on a P -algebra A isa linear map DA W A ! A such that for any operation � 2 P .n/ the followingformula holds:

DA.�.a1; : : : ; an// DnX

iD1�.a1; : : : ;D

A.ai /; : : : ; an/:

A similar computation as in the previous sections shows that the operad governingP -algebras with derivation, denoted by P Der, is such that

P Der.n/ D KŒx1; : : : ; xn�˝P .n/

the action of the symmetric group Sn being the diagonal action (recall that Snis acting on KŒx1; : : : ; xn� by permuting the variables). The composition � isobtained by combining the composition in P and the formula in Theorem 6.2.

For instance ComDer.n/ D KŒx1; : : : ; xn� is an Sn-module and the composi-tion map � of the operad ComDer is given by the same formula as for the operad

18 J.-L. Loday

AsDer. It means that these formulas are compatible with the symmetric group ac-tion. Another way of phrasing this result is the following. Consider the forgetfulfunctor which associates to a symmetric operad P a nonsymmetric operad eP suchthat eP n D P .n/. Then we have CComDer D AsDer.

If we start with a ns operad P , then P Der is a ns operad, where P Dern DKŒx1; : : : ; xn�˝Pn. In terms of standard constructions in the operad framework,it is the Hadamard product of P with AsDer.

6 The “pre-shuffle algebra” of associative algebras with derivation

If we replace the space of polynomials (or formal power series) by the space ofnoncommutative polynomials (or noncommutative formal power series), then wedo not get an operad anymore. However there is a variation of the notion of oper-ads which permits us to provide a similar treatment in this noncommutative frame-work, it is the notion of “pre-shuffle algebra” due to M. Ronco [17] .

6.1 Pre-shuffle algebra [17]

A pre-shuffle algebra is a family of vector spaces Pn equipped with compositionmaps

i W Pm ˝Pn ! PmCn�1; 1 � i � m;defined for n � 1;m � 1, which satisfy the following relations:

.� i �/ i�1Cj � D � i .� j �/;

for 1 � i � l; 1 � j � m. In other words, the difference with ns operads is thatwe keep only axiom I and we disregard Axiom II (see 1.1), therefore any algebraicns operad is a pre-shuffle algebra.

We denote by PerAsDer the pre-shuffle algebra which is generated by a unaryoperation D and a binary operation �, which satisfy the following relations:

8ˆ̂ˆ̂<

ˆ̂ˆ̂:

� 1 � D � 2 � ;D 1 � D � 1 D C � 2 D ;.˛ i D/ j � D .˛ j �/ i D ;.˛ i �/ jC1 D D .˛ j D/ i � ;

for any operation ˛ and i < j .Observe that the first relation is the associativity of �, the second relation is say-

ing that D is a derivation, the third and fourth relations say that the operations Dand � commute for parallel composition.


6.2 Theorem. As a vector space PerAsDern is isomorphic to the space of noncom-mutative polynomials in n variables:

PerAsDern D Khx1; : : : ; xni:

The composition map i is given by

.P iQ/.x1; : : : ; xnCm�1/D P.x1; : : : ; xi�1; xi C � � � C xiCm�1; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

Under this identification the operations id;D; � correspond to 11; x1 2 Khx1iand to 12 2 Khx1; x2i respectively. More generally, the operation

�� ..� jk D/ jk�1 D/ � � � j1 D

�

corresponds to the noncommutative monomial xjkxjk�1 � � � xj1 .

Graphically the operation xjkxjk�1 � � � xj1 is pictured as a planar decorated treewith levels as follows (example: x1x2xnx2) :

� � �

D � � �

� � � D

D � � �

D��

��

��

� � � ��

��

Proof. By [17] we know that the free preshuffle algebra on a certain set of generat-ing operations is spanned by some leveled planar trees whose vertices are labeledby the operations. Because of the relations entwining D and � we can move theoperations D up in a leveled tree composition of operations. Because of the asso-ciativity of � the trees involving only � give rise to corollas (as in the AsDer case).Hence the operad PerAsDer is spanned by the trees of the form indicated above.Note that the levels indicating the order in which the copies of the operationD areperformed is necessary in the preshuffle algebra framework.

6.3 Remark

One can also check that PerAsDer is a shuffle algebra in the sense of [17].

20 J.-L. Loday

6.4 Formal group laws in noncommutative variables

Let F.x; y/ be a formal group law in noncommutative variables. This is a seriesin K..x; y// which satisfies the relation

F.F.x; y/; z/ D F.x; F.y; z//:

For instance, the Baker-Campbell-Hausdorff series is the formal power series de-fined as

BCH.x; y/ WD log.exp.x/ exp.y//:

We recall that the first terms are

BCH.x; y/ D x C y C 12Œx; y�C 1

12.ŒŒx; y�; y�C Œx; Œx; y��/C � � � :

By induction we define

F.x1; : : : ; xn/ D F.F.x1; : : : ; xn�1/; xn/:

6.5 Proposition. For any formal group law F in noncommutative variables thereis a well-defined preshuffle algebra PerRatFctF given by:

PerRatFctFn WD K..x1; : : : ; xn//;

.P ıiQ/.x1; : : : ; xnCm�1/WD P.x1; : : : ; xi�1; F .xi ; � � � ; xiCm�1/; xiCm; : : : ; xnCm�1/�Q.xi ; : : : ; xiCm�1/:

If F is polynomial, then the restriction to Khx1; : : : ; xni is still a preshuffle alge-bra.

Proof. Axiom I is an immediate consequence of the equality

F.x1; : : : ; xi ; F .xiC1; : : : ; xiCj /; xiCjC1; : : : ; xn/ D F.x1; : : : ; xn/:

The last assertion is immediate.

6.6 Corollary. Let F be the additive formal group law F.x; y/ D x C y. Thenthe associated polynomial preshuffle algebra is PerAsDer.


7 Appendix: Homotopy associative algebras with derivation

We know that homotopy associative algebras are A1-algebras as defined by JimStasheff in [18]. Our purpose is to describe the notion of homotopy associative al-gebras with derivation, that is to unravel the operad AsDer1. The solution is givenby the Koszul duality theory of quadratic operads, see for instance [13] where theGinzburg-Kapranov theory is extended to operads generated by binary and unaryoperations. A quadratic operad P admits a Koszul dual cooperad P ¡. The cobarconstruction over P ¡ is the operad of P -algebras up to homotopy (i.e. the minimalmodel P1 WD �P ¡ of the operad P ) whenever the Koszul complex of the operadP is acyclic. In this appendix we compute the cooperad AsDer ¡ and its lineardual AsDerŠ, we prove that the Koszul complex .AsDer ¡ ıAsDer; ı/ is acyclic andwe unravel the co! bar construction AsDer1 WD �AsDer ¡. So we get a precisedescription of the notion of associative algebra with derivation up to homotopy.

If the parameter � 2 K is different from 0, then the operad �-AsDer is nota quadratic operad since the term D.a/D.b/ needs three generating operations tobe defined. So one needs new techniques to extend Koszul duality to this case,see [15].

7.1 The AsDerŠ-algebras

The relations defining the operad AsDer are quadratic since each monomial in-volves only the composition of two operations. Hence AsDer is suitable for apply-ing the Koszul duality theory. We use the notation and results of [13].

7.2 Proposition. The Koszul dual operad of AsDer is the operad AsDerŠ generatedby the unary operation d and the binary operation � which satisfy the followingrelations:

d ı d D 0; d ı � D � ı .d; id/ D � ı .id; d /; � ı .�; id/ D � ı .id; �/:

In other words an AsDerŠ-algebra is an associative algebra A equipped witha linear map d such that d2 D 0 and d.ab/ D d.a/b D ad.b/.

Proof. The operad AsDer is generated by the graded vector space

E D .0;KD;K�; 0; : : :/:

The weight 2 subspace of the free operad T .E/, denoted by T .E/.2/, is spannedby the operations which are composites of two of the generating operations. It isof dimension 6 with basis

� ı1 �; � ı2 �; � ı1 D; � ı2 D; D ı1 �; D ı1 D:

22 J.-L. Loday

The subspace of relationsR is of dimension 2 spanned byDı1��ı1D��ı2Dand � ı1 � � � ı2 �. The Koszul dual cooperad AsDer ¡ is cogenerated by sDand s� (s is the shift of degree), with s2R as corelations:

AsDer ¡ D Id˚ sE ˚ s2R˚ � � � :

By definition, the Koszul dual operad AsDerŠ of AsDer is, essentially, the lineardual of AsDer ¡. Let us denote by d the linear dual of sD (put in degree 0) andby � the linear dual of s� (put in degree 0). Then the space T .Kd ˚ K�/.2/is also of dimension 6 and the quotient AsDerŠ2 D R_ D T .Kd ˚ K�/=R? istwo-dimensional. So, R? is the 4-dimensional space spanned by the elements

d ı1 d; d ı1 � � � ı1 d; d ı1 � � � ı2 d; � ı1 � � � ı2 �:

7.3 Proposition. The space of n-ary operations of the operad AsDerŠ is two-dimensional:

AsDerŠn D K�n ˚K d�n; n � 2; and AsDerŠ1 D K id˚K d:

The partial composition ıi is given by

�m ıi �n D �mCn�1;d�m ıi �n D d�mCn�1;�m ıi d�n D d�mCn�1;d�m ıi d�n D 0;

where, by convention, �1 D id (so d ı1 �n D d�n).

Proof. T he generating binary operation � generates the operation �n in arity n.The relations entwining � and d imply that the only other possibility to create anoperation in arity n is to compose with a copy of d . The formula for the partialcomposition is obtained by direct inspection.

7.4 Proposition. The operad AsDer is a Koszul operad.

Proof. Most of the methods for proving Koszul duality would work in this simplecase. We choose to write down explicitly the “rewriting system method”, see [13].

The operad AsDer is presented by the generators D and � and the rewritingrelations ´

� ı1 � 7! � ı2 �;D ı1 � 7! � ı1 D C � ı2 D:


The critical monomials are � ı1 .� ı1 �/ and D ı1 .� ı1 �/. The first criticalmonomial is known to be confluent (Koszulity of the operad As), but let us recallthe proof. One one hand, one has

..xy/z/t 7! .x.yz//t 7! x..yz/t/ 7! x.y.zt//:

On the other hand, one has

..xy/z/t 7! .xy/.zt/ 7! x.y.zt//:

Since one ends up with the same element, we have shown that the first criticalmonomial is confluent.

Let us show confluence for the second critical monomial. One one hand, onehas

D..xy/z/ 7! D.x.yz// 7! .Dx/.yz/ 7! x.D.yz//7! .Dx/.yz/C x..Dy/z C y.Dz//7! .Dx/.yz/C x..Dy/z/C x.y.Dz//:

On the other hand, one has

D..xy/z/ 7! .D.xy//z/C .xy/.Dz/ 7! ..Dx/y/z C .x.Dy//z C .xy/.Dz/7! .Dx/.yz/C x..Dy/z/C x.y.Dz//:

Since one ends up with the same element, we are done.Since the critical monomials are confluent the ns operad AsDer is Koszul.

7.5 Homotopy associative algebra with derivation

Since the operad AsDer is a Koszul operad, its minimal model is given by theoperad �AsDer ¡. The aim of this section is to describe this operad explicitly.

By definition, a homotopy associative algebra with derivation is an algebra overthe differential graded ns operad AsDer1 constructed as follows. In arity n thespace .AsDer1/n is spanned by the planar trees with n leaves whose nodes with kinputs are labelled by mk or Dmk when k � 2, and by D when k D 1. Observethat the symbol Dmk is to be taken as a whole and not as mk followed by D. Thehomological degree ofmk and ofDmk is k�1 for k � 2, the homological degreeof D is 0.

24 J.-L. Loday

Example:

��

��

�

m2��

�� D��

Dm3

D

D

The differential map @ W .AsDer1/n ! .AsDer1/n of homological degree�1 is induced by

@.D/ D 0;

@.mn/ D �X

nDpCqCrkDpC1Crk>1;q>1

.�1/pCqrmk ı .id˝p ˝mq ˝ id˝r/;

@.Dmn/ D �X

nDpCqCrkDpC1Crk�1;q>1

.�1/pCqrDmk ı .id˝p ˝mq ˝ id˝r /

�X

nDpCqCrkDpC1Crk>1;q�1

.�1/pCqrmk ı .id˝p ˝Dmq ˝ id˝r/

for n � 2.For instance, in low arity we get:

@.m2/ D 0;@.m3/ D m2 ı .id; m2/ �m2 ı .m2; id/;

@.Dm2/ D �D ım2 Cm2 ı .id;D/Cm2 ı .D; id/:

7.6 Proposition. The operad AsDer1 is isomorphic to the minimal model�AsDer ¡ of the operad AsDer. Hence a homotopy associative algebra withderivation is an algebra over the operad AsDer1.

Proof. By definition the cobar construction over the cooperad AsDer ¡1 is the freeoperad T .sAsDer ¡/. Since, in arity n � 2, the space .AsDer ¡/n is spanned bytwo elements and the space AsDer ¡1 by one element (the notation overline means


that we get rid of the identity operation), the free operad T .sAsDer ¡/ is spannedby the planar rooted trees with labelled nodes as in the description of AsDer1. Inthe free operad the operadic composition is given by grafting.

The operad�AsDer ¡ is a differential graded operad, so we need to describe thedifferential map. It is sufficient to describe it on the operadic generators, that is onthe corollas labelled by eithermk orDmk . This boundary map is deduced from thecooperad structure of AsDer ¡, that is, from the operad structure of AsDerŠ. Fromthe formulas in Proposition 7.3 we deduce the formulas given in the constructionof the operad AsDer1.

7.7 Transfer theorem

Let us recall that the interest of the notion of “algebra up to homotopy” lies, in part,in the following transfer theorem. Let .A; ı/ be a differential graded associativealgebra with derivation DA of degree 0, that is ı.ab/ D ı.a/b C .�1/jajaı.b/and ı.DA.a// D DA.ı.a//, and let .V; ı/ be a retract by deformation of the chaincomplex .A; ı/ (e.g. .H.A/; 0/ when K is a field). Then .V; ı/ can be equippedwith a AsDer1-algebra structure transferred from the AsDer-algebra structure of.A; ı/. In particular there exist analogues of the Massey products on H.A/. SinceAsDer is Koszul, all these results are particular examples of [13], Chapter 9,which extend the results obtained by Tornike Kadeishvili [6] on differential gradedassociative algebras.

Bibliography

[1] F. Chapoton, Algèbres de Hopf des permutahedres, associahèdres et hypercubes(French) [Hopf algebras of permutahedra, associahedra and hypercubes], Adv. Math.150:2 (2000), 264–275. Note 2

Pleaseupdate [5,13, 17].

[2] F. Chapoton, The anticyclic operad of moulds, Int. Math. Res. Not. 2007:20, Art. IDrnm078, 36 pp.

[3] F. Chapoton, F. Hivert, J.-C. Novelli, and J.-Y. Thibon, An operational calculus forthe mould operad, Int. Math. Res. Not. 2008:9, Art. ID rnn018, 22 pp.

[4] V. Dotsenko, Compatible associative products and trees, J. Algebra Number Theory,3:5 (2009), 567–586.

[5] V. Dotsenko and A. Khoroshkin, Gröbner basis for operads, Duke Math. J. (to ap-pear).

[6] T. V. Kadeishvili, The algebraic structure in the homology of anA.1/-algebra (Rus-sian), Soobshch. Akad. Nauk Gruzin. SSR 108:2 (1982), 249–252 (1983).

26 J.-L. Loday

[7] Li Guo and W. Keigher, On differential Rota–Baxter algebras, J. Pure Appl. Algebra212:3 (2008), 522–540.

[8] J.-L. Loday, Algèbres ayant deux operations associatives (digèbres) (French) [Alge-bras with two associative operations (dialgebras)], C. R. Acad. Sci. Paris Sér. I Math.321:2 (1995), 141–146.

[9] J.-L. Loday, Dialgebras, in: Dialgebras and Related Operads, Lecture Notes inMath. 1763 pp. 7–66, Springer, Berlin, 2001.

[10] J.-L. Loday, Generalized bialgebras and triples of operads, Asterisque 320 (2008),x+116 pp.

[11] J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math.139:2 (1998), 293–309.

[12] J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in: HomotopyTheory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Contemp. Math. 2004 346, 369–398, Am. Math. Soc., Providence, RI.

[13] J.-L. Loday and B. Vallette, Algebraic operads, in preparation.

[14] M. Markl, J. Stasheff, and S. Shnider, Operads in Algebra, Topology and Physics,Mathematical Surveys and Monographs 96, Am. Math. Soc., Providence, RI, 2002.

[15] S. Merkulov and B. Vallette, Deformation theory of representations of prop(erad)s, I,J. Reine Angew. Math. 634 (2009), 51–106.

[16] T. Robinson, New perspectives on exponentiated derivations, the formal Taylor the-orem, and Faà di Bruno’s formula, arXiv:0903.3391[math].

[17] M. Ronco, Shuffle bialgebras, Ann. Inst. Fourier, 2010 (to appear).

[18] J. D. Stasheff, Homotopy associativity of H -spaces, I, II, Trans. Am. Math. Soc. 108(1963), 275–292; ibid. 108 (1963), 293–312.

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Author information

Jean-Louis Loday, Institut de Recherche Mathématique Avancée, CNRS et Université deStrasbourg, 7 rue R. Descartes, 67084 Strasbourg Cedex, France.E-mail: [email protected]

arXiv:0903.3391 [math]

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